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    "# 序列模型\n",
    "\n",
    "使用正弦函数和一些可加性噪声来生成序列数据，时间步为 $1, 2, \\ldots, 1000$"
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    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import torch\n",
    "from torch import nn\n",
    "from d2l import torch as d2l\n",
    "\n",
    "T = 1000\n",
    "time = torch.arange(1, T + 1, dtype=d2l.float32)\n",
    "x = torch.sin(0.01 * time) + torch.normal(0, 0.2, (T,))\n",
    "d2l.plot(time, [x], 'time', 'x', xlim=[1, 1000], figsize=(6, 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "将数据映射为数据对 $y_t = x_t$ 和 $\\mathbf{x}_t = [x_{t-\\tau}, \\ldots, x_{t-1}]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "origin_pos": 9,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "tau = 4\n",
    "features = torch.zeros((T - tau, tau))\n",
    "for i in range(tau):\n",
    "    features[:, i] = x[i: T - tau + i]\n",
    "labels = x[tau:].reshape((-1, 1))\n",
    "\n",
    "batch_size, n_train = 16, 600\n",
    "train_iter = d2l.load_array((features[:n_train], labels[:n_train]),\n",
    "                            batch_size, is_train=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "使用一个相当简单的结构：只是一个拥有两个全连接层的多层感知机"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "origin_pos": 12,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "def init_weights(m):\n",
    "    if type(m) == nn.Linear:\n",
    "        nn.init.xavier_uniform_(m.weight)\n",
    "\n",
    "def get_net():\n",
    "    net = nn.Sequential(nn.Linear(4, 10),\n",
    "                        nn.ReLU(),\n",
    "                        nn.Linear(10, 1))\n",
    "    net.apply(init_weights)\n",
    "    return net\n",
    "\n",
    "loss = nn.MSELoss()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "训练模型"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "origin_pos": 16,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss: 0.071921\n",
      "epoch 2, loss: 0.060791\n",
      "epoch 3, loss: 0.057015\n",
      "epoch 4, loss: 0.054417\n",
      "epoch 5, loss: 0.052864\n"
     ]
    }
   ],
   "source": [
    "def train(net, train_iter, loss, epochs, lr):\n",
    "    trainer = torch.optim.Adam(net.parameters(), lr)\n",
    "    for epoch in range(epochs):\n",
    "        for X, y in train_iter:\n",
    "            trainer.zero_grad()\n",
    "            l = loss(net(X), y)\n",
    "            l.backward()\n",
    "            trainer.step()\n",
    "        print(f'epoch {epoch + 1}, '\n",
    "              f'loss: {d2l.evaluate_loss(net, train_iter, loss):f}')\n",
    "\n",
    "net = get_net()\n",
    "train(net, train_iter, loss, 5, 0.01)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "模型预测下一个时间步"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "origin_pos": 19,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
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    "onestep_preds = net(features)\n",
    "d2l.plot([time, time[tau:]], [x.detach().numpy(), onestep_preds.detach().numpy()], 'time',\n",
    "         'x', legend=['data', '1-step preds'], xlim=[1, 1000], figsize=(6, 3))"
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   "source": [
    "进行多步预测"
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   "source": [
    "multistep_preds = torch.zeros(T)\n",
    "multistep_preds[: n_train + tau] = x[: n_train + tau]\n",
    "for i in range(n_train + tau, T):\n",
    "    multistep_preds[i] = net(\n",
    "        multistep_preds[i - tau:i].reshape((1, -1)))\n",
    "\n",
    "d2l.plot([time, time[tau:], time[n_train + tau:]],\n",
    "         [x.detach().numpy(), onestep_preds.detach().numpy(),\n",
    "          multistep_preds[n_train + tau:].detach().numpy()], 'time',\n",
    "         'x', legend=['data', '1-step preds', 'multistep preds'],\n",
    "         xlim=[1, 1000], figsize=(6, 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "更仔细地看一下$k$步预测"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "origin_pos": 28,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
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      "text/plain": [
       "<Figure size 432x216 with 1 Axes>"
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     },
     "metadata": {
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     "output_type": "display_data"
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   ],
   "source": [
    "max_steps = 64\n",
    "\n",
    "features = torch.zeros((T - tau - max_steps + 1, tau + max_steps))\n",
    "for i in range(tau):\n",
    "    features[:, i] = x[i: i + T - tau - max_steps + 1]\n",
    "\n",
    "for i in range(tau, tau + max_steps):\n",
    "    features[:, i] = net(features[:, i - tau:i]).reshape(-1)\n",
    "\n",
    "steps = (1, 4, 16, 64)\n",
    "d2l.plot([time[tau + i - 1: T - max_steps + i] for i in steps],\n",
    "         [features[:, (tau + i - 1)].detach().numpy() for i in steps], 'time', 'x',\n",
    "         legend=[f'{i}-step preds' for i in steps], xlim=[5, 1000],\n",
    "         figsize=(6, 3))"
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